Kern/Ralph
coLab meeting

Who am I?

• Postdoc at MSU with Gideon Bradburd

  • continuous space population genetics

• PhD with Scott Nuismer at UI (def June 2020)

  • developed phenotypic models of coevolution

    • studied coevolutionary maintenance of mutualism

  • methods to measure coevolution from trait data

  • studied diffusion limits of eco-evo models

Coevolution

The reciprocal evolutionary response of two species to a shared ecological interaction

(Image courtesy H. Toju)

Coevolution in space

When this process plays out in spatially structured populations, characteristic patterns can emerge

(Data courtesy Toju et al 2006)

Measuring coevolution

• Simple models predict bivariate distribution of traits

• Can use maximum likelihood to infer strength of coevolution (Week & Nuismer 2019)

• Caveat: spatially implicit

  • Ignores isolation by distance (IBD)

  • Which can confound inference

• Accounting for IBD may increase power to detect coevolution

Coevolution in continuous space

Courtesy Powell & Prior (2016)

Questions

• How do the spatial scales at which phenotypic patterns emerge depend on:

  • dispersal distances?

  • coevolutionary selection?

• How do spatial scales of local adaptation relate to spatial scales of phenotypic patterns?

• How can we use this information to identify loci involved in coevolution?

Approach

• Develop continuous space phenotypic model accounting for:

  • limited dispersal

  • random genetic drift

  • spatially homogeneous abiotic stabilizing selection

  • biotic coevolutionary selection

    • focus on host-parasite interactions

Why Host-Parasite Interactions?

• Ecologically important interaction

  • Thought to promote/maintain biodiversity

• Genetics are relatively well understood

  • Evidence for gene-for-gene mechanisms mediating Linum marginale & Melampsora lini coevolution (Dodds et al 2006)

  • Evidence for balancing selection Daphnia magna & Pasteuria ramosa (Bento et al 2017, Andras et al 2020)

• Has motivated studies in local adaptation

Outline of local (non-spatial) dynamics

• \(H=\) host, \(P=\) parasite

• Fitness \(m_H,m_P\) determined by traits \(z_H,z_P\)

• Selection: \(\mathrm{Cov}(m_H,z_H), \ \mathrm{Cov}(m_P,z_P)\)

• Mean trait dynamics given by

  • \(\frac{d}{dt}\bar z_H=\mathrm{Cov}(m_H,z_H)+\delta_H \\ \frac{d}{dt}\bar z_P=\mathrm{Cov}(m_P,z_P)+\delta_P\)

  • \(\delta_H,\delta_P\) are stochastic processes representing drift

Trait-matching/mismatching
fitness model

Coevolutionary selection

• \(\mathrm{Cov}(m,z)=G\beta\)

• \(G_H,G_P=\) additive genetic variances

• \(\beta_H,\beta_P=\) selection gradients

• \(\beta_H = -B_H(\bar z_P-\bar z_H) \\ \beta_P = \ B_P(\bar z_H-\bar z_P)\)

• \(B_H,B_P=\) strengths of coevolutionary selection

What it looks like

• But what about continuous space and limited dispersal?

Adding Space

• 2D space (\(x_1,x_2\))-coordinates

  • Assume offspring normally distributed around parents

  • \(\frac{\partial\bar z_S}{\partial t}=\color{red}{G_S\beta_S}+\color{blue}{\frac{\sigma_S^2}{2}\left(\frac{\partial^2\bar z_S}{\partial x_1^2}+\frac{\partial^2\bar z_S}{\partial x_2^2}\right)}+\color{green}{\delta_S} \\ S=H,P\)

  • \(\sigma_H,\sigma_P\propto\) expected dispersal distances

Abiotic stabilizing selection

• Need to add abiotic stabilizing selection to prevent runaway coevolution

• \(\beta_H = \color{green}{A_H(\theta_H-\bar z_H)}-\color{blue}{B_H(\bar z_P-\bar z_H)} \\ \beta_P = \color{green}{A_P(\theta_P-\bar z_P)}+\color{blue}{B_P(\bar z_H-\bar z_P)}\)

• \(A_H,A_P=\) strengths of abiotic stabilizing selection

• \(\theta_H,\theta_P=\) abiotic optimal trait values

Caveats

• Assumes:

  • weak coevolutionary selection (\(B_H,B_P\ll1\)),

  • spatially homogeneous selection strengths

  • spatially homogeneous abiotic environment

  • traits encoded by many additive small effect loci

  • constant population densities in space and time

  • constant additive genetic variances

What it looks like

• Can use theory of random fields to get spatial autocorrelation of trait values

Matérn correlation function

• \(\xi=\) characteristic length scale

Idea of characteristic length

• The geographical scale at which
  spatial patterns emerge

A Matérn random field

Spatial (intraspecific) covariance Functions

• \(C_S(h)\approx V_SM_1(h/\xi_S), \ S=H,P\)

  • \(V_H = 1/N_H\sigma_H^2(A_H-B_H), \\ V_P = 1/N_P\sigma_P^2(A_P+B_P)\)

  • \(N_H,N_P=\) effective densities

  • \(\xi_H= \frac{\sigma_H}{\sqrt{G_H(A_H-B_H)}}, \ \xi_P = \frac{\sigma_P}{\sqrt{G_P(A_P+B_P)}}\)

Effect of dispersal distance

• Increased \(\sigma\) leads to panmixia

Effect of coevolution

• Coevolutionary selection leads to opposing outcomes in each species

Questions

• How do the spatial scales at which phenotypic patterns emerge depend on:

  • dispersal distances? ‘check sign’

  • coevolutionary selection? ‘check sign’

• How do spatial scales of local adaptation relate to spatial scales of phenotypic patterns?

• How can we use this information to identify loci involved in coevolution?

Measuring Local Adapation

• We have fitness functions \(m_H(z_H,z_P),m_P(z_P,z_H)\)

• So we can calculate difference in fitness for

  • local interactions (home) vs
  • interactions at a spatial lag \(h\) (away)

• \(\Delta_H(h)=B_H[C_{HP}(h)-C_{HP}(0)]\)

• \(\Delta_P(h)=B_P[C_{HP}(0)-C_{HP}(h)]\)

  • \(C_{HP}(h)=\) interspecific cross-covariance

• spatial scale of LA = spatial scale of \(C_{HP}(h)\)

Spatial (interspecific) cross-covariance

• \(C_{HP}(h)=\) correlation of mean traits between species separated by lag \(h\)

• \(C_{HP}(h)\approx\frac{2}{\sigma_H^2\sigma_P^2}\left(\frac{G_HB_H}{N_P(A_P+B_P)}K_0(h/\xi_H)*M_1(h/\xi_P)\\-\frac{G_PB_P}{N_H(A_H-B_H)}K_0(h/\xi_P)*M_1(h/\xi_H)\right)\)

• \(K_0(h)=\) modified Bessel function of the second kind

• \(*=\) convolution

• Can investigate using numerical convolution

Local adapation

• Decreased fitness with spatial lag \(\implies\) LA

Local adaptation
as function of dispersal

• The shorter disperser tends to be locally adapted

Local adaptation
as function of coevolution

• The species experiencing weaker biotic selection tends to be locally adapted, but the signal is faint

Next Steps

1. Check model results using simulations relaxing key assumptions

2. Work toward statistical inference method for identifying loci mediating coevolutionary interactions in continuous space

• Tricky because continuous space can lead to IBD and spurious interspecific spatial correlations of allele frequencies

• How can we control for interspecific genetic linkage due to IBD?